There are various research works on semi-competing risks data analysis in the literature. An overlapping principal consideration in works related to this field is to figure out how to deal with the dependence between the events of interest. A primitive method is to model time to the events of interest through two marginal distributions where there is no constraint on their dependence structure [1]}. The other popular method is to step forward and utilize copula to account for the dependence [2]}, [3]}. Another technique uses the conditional modeling approach to model the transition-specific hazard functions for the terminal and non-terminal events. The innovative work by [4]} is an example of this approach in which a frailty model for semi-competing risks data has been proposed. Furthermore, the multiplicative Cox model was employed for the corresponding three transitions. Also, the Gamma frailty and the non-parametric maximum likelihood estimation were utilized to manipulate the estimation task. Recently, [5]} extended the shared-frailty illness death model to right-censored and left-truncated semi-competing risks data under the multi-state modeling approach. The illness-death model is a simple non-trivial example of multi-state models in which individuals may undergo a transient (diseased) state before reaching a terminal (dead) state [6]}.
